Chromebook..
No, don't get it. It is not worth it.
I was intended to get one, but after some research, it was a waste.
Chromebook is similar to those 'Android' netbook. Only different is the name.
Android netbook is very much similar to android tablet.
Usage; internet (thru gmail account), android apps:- May not be compatible to Windows.
At your budget, just a mid-range laptop will do, maybe add RAM to 8GB (normally come with 4GB)
Nowadays laptop is very cheap. Also need to look at the specific spec.
My this laptop is Toshiba (Model Satellite L735; 2nd hand, maybe 2 years old) intel core i3 (i5, i7, now i9) with 500GB harddisk, and topup to 8GB ram. Also with USB3 (not many model will have).
Software; Windows 7 Premium; MS Office (starter version), all kind of Toshiba software.
As for Game, look at the graphic card (more Ram the better, for fast action game), 64bit faster (mine is 32bit).
Lastly, if you are not familiar on the spec, then choose the common model.
Look for those common model basic (don't do your own customize) with intel core i3 (or i5) and graphic (with more ram).
This is because the laptop cannot do upgrade hardware, just like desktop can.
Harddisk can replace, later (if the spec mention 320GB).
Upgrade the RAM at the point of buying, so that you do not need to upgrade later.
Otherwise you may have problem, not available RAM slot. The std is 4GB = 2 pcs of 2GB on both RAM slot (std 2 slots).
Hope you understand......
edit: Screen size. 13.3" to 15" depend on your budget.
Don't worry about screen size, you can get 21" TV with RGB connection. Most Laptop do have RGB connection.
10" - will not have DVD drive, with many feature not available. I had one before.
12" - i am not too sure
Mine is 13.3". Just alright for me. My previous is 14". Both have DVD drive.